number theory

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Mahfuz Sobhan
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Joined: Sat Feb 07, 2015 5:40 pm

number theory

Unread post by Mahfuz Sobhan » Wed Jul 15, 2015 12:19 pm

Let us define $$S_n$$ to be the set of all integers divisible by $$2014^n$$ but not $$2014^n+^1$$ where $$n$$ is a non-negative integer. What is the value of $$n$$ (if any) so that $$500!$$ belongs to $$S_n$$. :?:

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Re: number theory

Unread post by seemanta001 » Mon Jul 20, 2015 1:08 am

We can observe that $$2014=2\times19\times53$$.Here $53$ is the biggest prime factor of $2014$.
We have to find $\sum\frac{500}{53^n}$ for $n>0$.
There are $9$ numbers that are multiples of $53$ from $1$ to $500$.
No other multiples of $53^n$ are within numbers $1$ to $500$,where $n>1$.
Thus,we get our answer.
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